Sol. The magnetic field lines inside the toroid are circles concentric with the toroid. We choose a circle of radius r inside the solenoid to perform integration of Ampere’s law. We make this choice of the symmetry of the situation. B is same at all points along the path and tangential. So from Ampere’s law.

Where N is the total number of coils and I is the current in each of the coils. Thus

The magnitude of field is not constant over the cross-sectional area: it is largest along the inner edge. For a large and thin toroid, this difference can be negligible; the field will be uniform.

Outside the toroid we choose as our closed curve a circle concentric with the toroid. This closed curve encloses N loops carrying current I in one direction and N loops carrying same current in opposite direction. Thus the net current enclosed is zero. From ampere’s law,

= μ 0 I encl

B(2 π r) = 0

B = 0

The same is true for a circle with radius smaller than that of the toroid.